MinT - Best Known (129, 129+64, s)-Nets in Base 4

Best Known (129, 129+64, s)-Nets in Base 4

(129, 129+64, 195)-Net over F₄ — Constructive and digital

Digital (129, 193, 195)-net over F₄, using

4¹ times duplication [i] based on digital (128, 192, 195)-net over F₄, using
- trace code for nets [i] based on digital (0, 64, 65)-net over F₆₄, using
  - net from sequence [i] based on digital (0, 64)-sequence over F₆₄, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 0 and N(F) ≥ 65, using
      - the rational function field F₆₄(x) [i]
    - Niederreiter sequence [i]

(129, 129+64, 240)-Net in Base 4 — Constructive

(129, 193, 240)-net in base 4, using

3 times m-reduction [i] based on (129, 196, 240)-net in base 4, using
- trace code for nets [i] based on (31, 98, 120)-net in base 16, using
  - 2 times m-reduction [i] based on (31, 100, 120)-net in base 16, using
    - base change [i] based on digital (11, 80, 120)-net over F₃₂, using
      - net from sequence [i] based on digital (11, 119)-sequence over F₃₂, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 11 and N(F) ≥ 120, using
        a function field by SÃ©mirat [i]

(129, 129+64, 543)-Net over F₄ — Digital

Digital (129, 193, 543)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(129, 129+64, 18210)-Net in Base 4 — Upper bound on s

There is no (129, 193, 18211)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 157 882198 881321 667567 342425 197996 947959 735620 337053 150753 261405 812298 968198 772758 415025 896068 262370 024565 325550 667225 > 4¹⁹³ [i]